DFG Graduiertenkolleg 1821 "Cohomological Methods in Geometry"

The interplay between concrete geometry and abstract algebra is one of the driving principles of mathematics. Geometric concepts allow visualization and invite intuitive approaches. Algebraic methods make these amenable to computation. The bridge between algebra and geometry is used in both directions: geometric objects are characterized by numbers; algebraic objects are given geometric interpretations. We make this bridge the Leitmotiv of our proposal. Arguably the most prominent use of this principle is cohomology: it was introduced as a systematic way of defining invariants of geometric objects depending only on the shape, rather than on finer structures like distances and angles. The first example is the Euler number: for any polyhedron that "looks like a sphere" the alternating sum of the numbers of faces, edges and vertices is 2. It was later recognized that these invariants also have meaning in many other contexts. Dynamical systems on the sphere have 2 critical points, if counted with multiplicities - and it is no coincidence that the two numbers agree. From these beginnings, the technique was strengthened and generalized. It is now a very powerful tool central to all geometric disciplines. This allows the transfer of ideas and results ofrom one setting to another, which are prima facie completely unrelated. By bringing together strong representatives from the fields of number theory, algebraic geometry, differential geometry, representation theory and mathematical physics, our group covers a wide range of subjects. Our methods are, however, closely related. Cohomology has already been mentioned. In its many incarnations it appears throughout all our research topics. Other techniques used in our projects are Hodge theory, deformation theory, Lie groups and algebraic geometry.

Weitere Informationen: http://www.gk1821.uni-freiburg.de/
Ansprechpartner: Prof. Dr. Annette Huber-Klawitter
Tel: +49 761 203-5560
Email: annette.huber@math.uni-freiburg.de

Projektbeginn: 01.10.2012
Projektende: 30.09.2021

Prof. Dr. Annette Huber-Klawitter
Stellvertretung: Prof. Dr. S. Kebekus
Albert-Ludwigs-Universität Freiburg

• DFG